The Euler number of the Jacobi factor of a plane curve singularity whose semigroup is $⟨4,6,13⟩$

Piontkowski calculated the Euler number of Jacobi factors of plane curve singularities with semigroups $<p,q>$, $<4,2q,s>$, $<6,8,s>$ and $< 6,10, s>$. %His analysis was done by decomposing the Jacobi factors into affine cells. In this paper, we show that a Jacobi factor for any curve singularity admits a cell decomposition by virtue of Pfister and Steenbrink’s theory for punctual Hilbert schemes. We also introduce a computational method to determine the number of affine cells in the decomposition. Applying it, we compute the the Euler number of the Jacobi factor of a singularity with a semigroup $< 4,6,13>$. Our result gives a counterexample for Piontkowski’s calculation.